Quantum Markov states and quantum hidden Markov states

The present article is a continuation of our previous paper “Remarks on quantum Markov states” (Funct. Anal. Appl. 49, No. 3 (2015), 205–209). We prove some propositions from that paper and define quantum Markov states and quantum hidden Markov states. Some connections are established with other definitions of these notions. We consider such states for lattices and graphs. We also consider an example with the Cayley tree.